# Course Descriptions

MATH 521
Algebra I
Credits4
Free groups, group actions, group with operators, Sylow theorems, Jordan-Hölder theorem, nilpotent and solvable groups. Polynomial and power series rings, Gauss's lemma, PID and UFD, localization and local rings,chain conditions, Jacobson radical.
Prerequisite: None
MATH 522
Algebra II
Credits4
Galois theory, solvability of equations by radicals, separable extensions, normal basis theorem, norm and trace, cyclic and cyclotomic extensions, Kummer extensions. Modules, direct sums, free modules, sums and products, exact sequences, morphisms, Hom and tensor functors, duality, projective, injective and flat modules, simplicity and semisimplicity, density theorem, Wedderburn-Artin theorem, finitely generated modules over a principal ideal domain, basis theorem for finite abelian groups.
Prerequisite: None
MATH 525
Algebraic Number Theory
Credits4
Valuations of a field, local fields, ramification index and degree, places of global fields, theory of divisors, ideal theory, adeles and ideles, Minkowski's theory, extensions of global fields, the Artin symbol.
Prerequisite: None
MATH 527
Number Theory
Credits4
Method of descent, unique factorization, basic algebraic number theory, diophantine equations, elliptic equations, p-adic numbers, Riemann zeta function, elliptic curves, modular forms, zeta and L-functions, ABC-conjecture, heights, class numbers for quadratic fields, a sketch of Wiles' proof.
Prerequisite: None
MATH 528
Analytic Number Theory
Credits4
Primes in arithmetic progressions, Gauss' sum, primitive characters, class number formula, distribution of primes, properties of the Riemann zeta function and Dirichlet L-functions, the prime number theorem, Polya- Vinogradov inequality, the large sieve, average results on the distribution of primes.
Prerequisite: Math 533
MATH 529
Analytic Number Theory II
Credits3
The prime number theorem for arithmetic progressions. Sums over primes, exponential sums. The large sieve, Bombieri-Vinogradov theorem,Selberg’s sieve. Results on the distribution of primes.
Prerequisite: Math 528
MATH 531
Real Analysis I
Credits4
Lebesgue measure and Lebesgue integration on Rn, general measure and integration, decomposition of measures, Radon-Nikodym theorem, extension of measures, Fubini's theorem.
Prerequisite: None
MATH 532
Real Analysis II
Credits4
Normed and Banach spaces, Lp-spaces and duals, Hahn-Banach theorem, category and uniform boundedness theorem, strong, weak and weak*-convergence, open mapping theorem, closed graph theorem.
Prerequisite: Math 531
MATH 533
Complex Analysis I
Credits4
Review of the complex number system and the topology of C, elementary properties and examples of analytic functions, complex integration, singularities, maximum modulus theorem, compactness and convergence in the space of analytic functions.
Prerequisite: None
MATH 534
Complex Analysis II
Credits4
Runge's theorem, analytic continuation, Riemann surfaces, harmonic functions, entire functions, the range of an analytic function.
Prerequisite: MATH 533
MATH 535
Functional Analysis
Credits4
Topological vector spaces, locally convex spaces, weak and weak* topologies, duality, Alaoglu's theorem, Krein-Milman theorem and applications, Schauder fixed point theorem, Krein-Smulian theorem, Eberlein-Smulian theorem, linear operators on Banach spaces.
Prerequisite: MATH 531 and MATH 532
MATH 541
Probability Theory
Credits4
An introduction to measure theory, Kolmogorov axioms, independence, random variables, expectation, modes of convergence for sequences of random variables, moments of a random variable, generating functions, characteristic functions, product measures and joint probability, distribution laws, conditional expectations, strong and weak law of large numbers, convergence theorems for probability measures, central limit theorems.
Prerequisite: None
MATH 544
Stochastic Processes and Martingales
Credits4
Stochastic processes, stopping times, Doob-Meyer decomposition, Doob's martingale convergence theorem, characterization of square integrable martingales, Radon-Nikodym theorem, Brownian motion, reflection principle, law of iterated logarithms.
Prerequisite: MATH 541
MATH 545
Mathematics of Finance
Credits4
From random walk to Brownian motion, quadratic variation and volatility, stochastic integrals, martingale property, Ito formula, geometric Brownian motion, solution of Black-Scholes equation, stochastic differential equations, Feynman-Kac theorem, Cox-Ingersoll-Ross and Vasicek term structure models, Girsanov's theorem and risk neutral measures, Heath-Jarrow-Morton term structure model, exchange-rate instruments.
Prerequisite: None
MATH 551
Partial Differential Equations I
Credits4
Existence and uniqueness theorems for ordinary differential equations, continuous dependence on data. Basic linear partial differential equations : transport equation, Laplace's equation, diffusion equation, wave equation. Method of characteristics for non-linear first-order PDE's, conservation laws, special solutions of PDE's, Cauchy-Kowalevskaya theorem.
Prerequisite: None
MATH 552
Partial Differential Equations II
Credits4
Hölder spaces, Sobolev spaces, Sobolev embedding theorems, existence and regularity for second-order elliptic equations, maximum principles, second-order linear parabolic and hyperbolic equations, methods for non-linear PDE's, variational methods, fixed point theorems of Banach and Schauder.
Prerequisite: MATH 551
MATH 571
Topology
Credits4
Fundamental concepts, subbasis, neighborhoods, continuous functions, subspaces, product spaces and quotient spaces, weak topologies and embedding theorem, convergence by nets and filters, separation and countability, compactness, local compactness and compactifications, paracompactness, metrization, complete metric spaces and Baire category theorem, connectedness.
Prerequisite: None
MATH 572
Algebraic Topology
Credits4
Basic notions on categories and functors, the fundamental group, homotopy, covering spaces, the universal covering space, covering transformations, simplicial complexes and their homology.
Prerequisite: MATH 571
MATH 575
Differentiable Manifolds
Credits3
Differentiable manifolds, smooth maps, submanifolds, vectors and vector fields, Lie brackets, Lie Groups, Lie group actions, integral curves and flows, Lie algebras, Lie derivative, Killing fields, differential forms, Integration.
Prerequisite: None
MATH 576
Riemannian Geometry
Credits3
Differentiable manifolds, vectors and tensors, riemannian metrics, connections, geodesics, curvature, jacobi fields, riemannian submanifolds, spaces of constant curvature.
Prerequisite: None
MATH 577
Complex Manifolds
Credits3
Complex Manifolds, Kahler and Calabi-Yau Manifolds, Homology and Cohomology, Fiber Bundles, Connections on Fiber Bundles, Characteristic Classes, Index Theorems.
Prerequisite: MATH 576 or consent of the instructor
MATH 579
Credits0
Presentation of topics of interest in mathematics through seminars offered by faculty, guest speakers and graduate students.
Prerequisite: None
MATH 581
Selected Topics in Analysis I
Credits3
MATH 582
Selected Topics in Analysis II
Credits3
MATH 583
Selected Topics in Foundations of Mathematics
Credits3
MATH 584
Selected Topics in Algebra and Topology
Credits3
MATH 585
Selected Topics in Probability and Statistics
Credits3
MATH 586
Selected Topics in Differential Geometry
Credits3
MATH 587
Selected Topics in Differential Equations
Credits3
MATH 588
Selected Topics in Applied Mathematics I
Credits3
MATH 589
Selected Topics in Combinatorics
Credits3
MATH 590
Credits1
Literature survey and presentation on a subject to be determined by the instructor.
Prerequisite: None
MATH 601
Measure Theory
Credits4
Fundamentals of measure and integration theory, Radon-Nikodym Theorem, Lp spaces, modes of convergence, product measures and integration over locally compact topological spaces.
Prerequisite: None
MATH 611
Differential Geometry I
Credits4
Survey of differentiable manifolds, Lie groups and fibre bundles, theory of connections, holonomy groups, extension and reduction theorems, applications to linear and affine connections, curvature, torsion, geodesics, applications to Riemannian connections, metric normal coordinates, completeness, De Rham decomposition theorem, sectional curvature, spaces of constant curvature, equivalence problem for affine and Riemannian connection.
Prerequisite: None
MATH 612
Differential Geometry II
Credits4
Submanifolds, fundamental theorem for hypersurfaces, variations of the length integral, Jacobi fields, comparison theorem, Morse index theorem, almost complex and complex manifolds, Hermitian and Kaehlerian metrics, homogeneous spaces, symmetric spaces and symmetric Lie algebra, characteristic classes.
Prerequisite: MATH 611
MATH 623
Integral Transforms
Credits4
Fourier transforms, exponential, cosine and sine, Fourier transform in many variables, application of Fourier transform to solve boundary value problems, Laplace transform, use of residue theorem and contour integration for the inverse of Laplace transform, application of Laplace transform to solve differential and integral equations, Fourier-Bessel and Hankel transforms for circular regions, Abel transform for dual integral equations.
Prerequisite: None
MATH 624
Numerical Solutions of Partial Differential and Integral Equations
Credits4
Parabolic differential equations, explicit and implicit formulas, elliptic equations, hyperbolic systems, finite elements characteristics, Volterra and Fredholm integral equations.
Prerequisite: None
MATH 627
Optimization Theory I
Credits4
Fundamentals of linear and nonlinear optimization theory. Unconstrained optimization, constrained optimization, saddlepoint conditions, Kuhn-Tucker conditions, post-optimality, duality, convexity, quadratic programming, multistage optimization.
Prerequisite: None
MATH 628
Optimization Theory II
Credits4
Design and analysis of algorithms for linear and non-linear optimization. The revised simplex method, algorithms for network problems, dynamic programming, quadratic programming techniques, methods for constrained nonlinear problems.
Prerequisite: MATH 627
MATH 631
Algebraic Topology I
Credits4
Basic notions on categories and functions, the fundamental groups, homotopy, covering spaces, the universal covering space, covering transformations, simplicial complexes and homology of simplicial complexes.
Prerequisite: None
MATH 632
Algebraic Topology II
Credits4
Singular homology, exact sequences, the Mayer-Vietoris exact sequence, the Lefschetz fixed-point theorem, cohomology, cup and cap products, duality theorems, the Hurewicz theorem, higher homotopy groups.
Prerequisite: None
MATH 635
An Introduction to Nonlinear Analysis
Credits3
Calculus in Banach spaces. Implicit function theorems. Degree theories. Fixed Point Theorems. Bifurcation theory. Morse Lemma. Variational methods. Critical points of functionals. Palais-Smale condition. Mountain Pass Theorem.
Prerequisite: MATH 535 or equivalent
MATH 643
Stochastic Processes I
Credits4
Survey of measure and integration theory, measurable functions and random variables, expectation of random variables, convergence concepts, conditional expectation, stochastic processes with emphasis on Wiener processes, Markov processes and martingales, spectral representation of second-order processes, linear prediction and filtering, Ito and Saratonovich integrals, Ito calculus, stochastic differential equations, diffusion processes, Gaussian measures, recursive estimation.
Prerequisite: MATH 552 or consent of instructor.
MATH 644
Stochastic Processes II
Credits4
Tightness, Prohorov's theorem, existence of Brownian motion, Martingale characterization of Brownian motion, Girsanov's theorem, Feynmann-Kac formulas, Martingale problem of Stroock and Varadhan, applications to mathematics of finance.
Prerequisite: MATH 643
MATH 645
Mathematical Statistics
Credits4
Review of essentials of probability theory, subjective probability and utility theory, statistical decision problems, a comparison game theory and decision theory, main theorems of decision theory with emphasis on Bayes and minimax decision rules, distribution and sufficient statistics, invariant statistical decision problem, testing hypotheses, the Neyman-Pearson lemma, sequential decision problem.
Prerequisite: MATH 552 or consent of instructor.
MATH 660
Credits4
Basic algebraic number theory; number fields, ramification theory, class groups, Dirichlet unit theorem; zeta and L-functions; Riemann, Dedekind zeta functions, Dirichlet, Hecke L-functions, primes in arithmetic progressions, prime number theorem; cyclotomic fields, reciprocity laws, class field theory, ideles and adeles, modular functions and modular forms.
Prerequisite: None
MATH 680
Seminar in Pure Mathematics I
Credits4
Recent developments in pure mathematics.
Prerequisite: None
MATH 681
Seminar in Pure Mathematics II
Credits4
Recent developments in pure mathematics.
Prerequisite: None
MATH 682
Seminar in Applied Mathematics I
Credits4
Recent developments in applied mathematics.
Prerequisite: None
MATH 683
Seminar in Applied Mathematics II
Credits4
Recent developments in applied mathematics.
Prerequisite: None
MATH 690
M.S. Thesis
MATH 699
Guided Research
Credits4
Research in the field of Mathematics, by arrangement with members of the faculty; guidance of doctoral students towards the preparation and presentation of a research proposal.
Prerequisite: None
MATH 790
Ph.D. Thesis